$\lim_{x\to\infty}\dfrac{6x}{15x-8}=?$ Choose 1 answer: Choose 1 answer: (Choice A) A $0$ (Choice B) B $0.4$ (Choice C) C $2.5$ (Choice D) D $\infty$
Solution: $\lim_{x\to\infty} 6x=\infty$ and $\lim_{x\to\infty} 15x-8=\infty$, so $\lim_{x\to\infty}\dfrac{6x}{15x-8}$ results in the indeterminate form $\dfrac{\infty}{\infty}$. We should use l'Hôpital's rule. $\begin{aligned} &\phantom{=}\lim_{x\to\infty}\dfrac{6x}{15x-8} \\\\ &=\lim_{x\to\infty}\dfrac{\dfrac{d}{dx}\left[6x\right]}{\dfrac{d}{dx}[15x-8]} \gray{\text{l'Hôpital's rule}} \\\\ &=\lim_{x\to\infty}\dfrac{6}{15} \\\\ &=0.4 \end{aligned}$ Note that we were only able to use l'Hôpital's rule because the limit $\lim_{x\to\infty}\dfrac{\dfrac{d}{dx}\left[6x\right]}{\dfrac{d}{dx}[15x-8]}$ can actually be determined. In conclusion, $\lim_{x\to\infty}\dfrac{6x}{15x-8}=0.4$.